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On topological entropy: When positivity implies +infinity


Authors: Sergiǐ Kolyada and Julia Semikina
Journal: Proc. Amer. Math. Soc. 143 (2015), 1545-1558
MSC (2010): Primary 37B40; Secondary 54H20, 54H15
DOI: https://doi.org/10.1090/S0002-9939-2014-12282-9
Published electronically: December 4, 2014
MathSciNet review: 3314068
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Abstract: In this paper we study the relations between the properties of the topological semigroup of all continuous selfmaps $S(X)$ on a compact metric space $X$ (the topological group $H(X)$ of all homeomorphisms on $X$) and possible values of the topological entropy of its elements. In particular, we prove that topological entropy of a functional envelope on the space of all continuous selfmaps on Peano continua or on compact metric spaces with continuum many connected components has only two possible values $0$ and $+\infty$.


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Additional Information

Sergiǐ Kolyada
Affiliation: Institute of Mathematics, NASU, Tereshchenkivs’ka 3, 01601 Kyiv, Ukraine
MR Author ID: 196872
Email: skolyada@imath.kiev.ua

Julia Semikina
Affiliation: Mathematical Institute of the University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany
Email: julia.semikina@gmail.com

Received by editor(s): March 9, 2013
Received by editor(s) in revised form: June 2, 2013
Published electronically: December 4, 2014
Additional Notes: The first author was supported by Max-Planck-Institut für Mathematik (Bonn); he acknowledges the hospitality of the Institute
The second author was supported by Bonn International Graduate School in Mathematics
Communicated by: Yingfei Yi
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.